course Phy 231 I have completed assignment 1.Thanks ??????????O?G?
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21:05:12 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). The evaluate each expression for x = 2.
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RESPONSE --> The difference between the 2 expressions is that the first does not contain parenthesis, and therefore the order of operations is different. The resulting answers are: 1) 5 2) 0. confidence assessment: 3
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21:07:22 The order of operations dictates that grouped expressions must be evaluated first, that exponentiation must be done before multiplication or division, which must be done before addition or subtraction. It makes a big difference whether you subtract the 2 from the x or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract. Substituting 2 for x we get 2 - 2 / 2 + 4 = 2 - 1 + 4 (do multiplications and divisions before additions and subtractions) = 5 (add and subtract in indicated order) If there are parentheses you evaluate the grouped expressions first: (x - 2) / (x - 4) = (2 - 2) / ( 4 - 2) = 0 / 2 = 0.
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RESPONSE --> Agreed - I got the same answers, though my initial explanation was not as detailed. I can provide more in-depth critique of my answers going forward. self critique assessment: 3
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21:11:52 `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2. Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation.
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RESPONSE --> As in question 001, the difference lies in the parenthesis. While the 2nd expression has the parenthesis around the x+4, the first does not. This means that in the first expression, the only part that is exponentiated is the x; the 4 is then added to the 2^x expression. In the 2nd expression, however, the parenthesis ensures the 4 is added to the x, prior to exponentiating; this means the entire expression x+4 is exponentiated to the 2. In evaluating for x=2, the first expression is equal to 8, while the 2nd is equal to 64. confidence assessment: 3
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21:12:14 2 ^ x + 4 indicates that you are to raise 2 to the x power before adding the 4. 2 ^ (x + 4) indicates that you are to first evaluate x + 4, then raise 2 to this power. If x = 2, then 2 ^ x + 4 = 2 ^ 2 + 4 = 2 * 2 + 4 = 4 + 4 = 8. and 2 ^ (x + 4) = 2 ^ (2 + 4) = 2 ^ 6 = 2*2*2*2*2*2 = 64.
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RESPONSE --> Agreed. No self-critique needed self critique assessment: 3
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21:14:00 01-26-2009 21:14:00 `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2?
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NOTES -------> The only part of the expression that is a fraction is
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21:19:52 `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2?
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RESPONSE --> The only part of the expression that is a fraction is: 3 / [ (2x-5)^2 * 3x + 1 ] therefore, the numerator is 3, and the denominator is [ (2x-5)^2 * 3x + 1 ] ***I accidentally clicked ""Save as Notes"" and the question disappeared. I believe it stated to evaluate for x=2, therefore.... at x=2, the expression evaluates to: 2 - 3 / [(2(2) - 5) ^ 2 * 3(2) - 1] - 2 + 7(2) which equals 13 and 2/5 or 13.4 confidence assessment: 2
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21:22:31 The numerator is 3. x isn't part of the fraction. / indicates division, which must always precede subtraction. Only the 3 is divided by [ (2x-5)^2 * 3x + 1 ] and only [ (2x-5)^2 * 3x + 1 ] divides 3. If we mean (x - 3) / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x we have to write it that way. The preceding comments show that the denominator is [ (2x-5)^2 * 3x + 1 ] Evaluating the expression for x = 2: - 3 / [ (2 * 2 - 5)^2 * 3(2) + 1 ] - 2 + 7*2 = 2 - 3 / [ (4 - 5)^2 * 6 + 1 ] - 2 + 14 = evaluate in parenthese; do multiplications outside parentheses 2 - 3 / [ (-1)^2 * 6 + 1 ] -2 + 14 = add inside parentheses 2 - 3 / [ 1 * 6 + 1 ] - 2 + 14 = exponentiate in bracketed term; 2 - 3 / 7 - 2 + 14 = evaluate in brackets 13 4/7 or 95/7 or about 13.57 add and subtract in order. The details of the calculation 2 - 3 / 7 - 2 + 14: Since multiplication precedes addition or subtraction the 3/7 must be done first, making 3/7 a fraction. Changing the order of the terms we have 2 - 2 + 14 - 3 / 7 = 14 - 3/7 = 98/7 - 3/7 = 95/7. COMMON STUDENT QUESTION: ok, I dont understand why x isnt part of the fraction? And I dont understand why only the brackets are divided by 3..why not the rest of the equation? INSTRUCTOR RESPONSE: Different situations give us different algebraic expressions; the situation dictates the form of the expression. If the above expression was was written otherwise it would be a completely different expression and most likely give you a different result when you substitute. If we intended the numerator to be x - 3 then the expression would be written (x - 3) / [(2x-5)^2 * 3x + 1 ] - 2 + 7x, with the x - 3 grouped. If we intended the numerator to be the entire expression after the / the expression would be written x - 3 / [(2x-5)^2 * 3x + 1 - 2 + 7x ].
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RESPONSE --> In the denominator, I had written down in my notes a "" - 1"", rather than a "" +1"", which resulted in a miscalculation of the denominator. I calculated the denominator as 5, when actually, it should have been 7. Were it not for this transcription error, I would have correctly solved the problem. self critique assessment: 2
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21:29:23 `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.
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RESPONSE --> because x-5 is in parenthesis, that is evaluated first. (X-5) is then raised to the 2x power; note: it is NOT raised to the 2x-1 power, because the 2x-1 is not in parenthesis. At x=4, (x-5)^2x power is equal to -1^8, which is equal to 1. Moving to the second portion of the equation, we divide 3 by x; again, note: NOT 3 divided by x-2, because the x-2 is not in parenthesis. At x=4, 3/x is equal to 3/4 (or .75). Putting the whole equation together, we have: 1 - 1 + 3/4 - 2, which equals -1 1/4 or -1.25. confidence assessment: 3
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21:32:19 We get (4-5)^2 * 4 - 1 + 3 / 1 - 4 = (-1)^2 * 4 - 1 + 3 / 4 - 2 evaluating the term in parentheses = 1 * 4 - 1 + 3 / 4 - 2 exponentiating (2 is the exponent, which is applied to -1 rather than multiplying the 2 by 4 = 4 - 1 + 3/4 - 2 noting that 3/4 is a fraction and adding and subtracting in order we get = 1 3/4 = 7 /4 (Note that we could group the expression as 4 - 1 - 2 + 3/4 = 1 + 3/4 = 1 3/4 = 7/4). COMMON ERROR: (4 - 5) ^ 2*4 - 1 + 3 / 4 - 2 = -1 ^ 2*4 - 1 + 3 / 4-2 = -1 ^ 8 -1 + 3 / 4 - 2. INSTRUCTOR COMMENTS: There are two errors here. In the second step you can't multiply 2 * 4 because you have (-1)^2, which must be done first.?Exponentiation precedes multiplication. ? Also it isn't quite correct to write -1^2*4 at the beginning of the second step. If you were supposed to multiply 2 * 4 the expression would be (-1)^(2 * 4).? Note also that the -1 needs to be grouped because the entire expression (-1) is taken to the power.?-1^8 would be -1 because you would raise 1 to the power 8 before applying the - sign, which is effectively a multiplication by -1.?......!!!!!!!!................................... RESPONSE --> I made a common error of sandwiching the 2 and the x from the exponentiation in the first part of the equation. The assumption on my part was made because of the proximity of the 2x, though it should have been known that this implies multiplication, which comes after exponentiation in the order of events. self critique assessment: 2
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21:34:36 *&*& Standard mathematics notation is easier to see. On the other hand it's very important to understand order of operations, and students do get used to this way of doing it. You should of course write everything out in standard notation when you work it on paper. It is likely that you will at some point use a computer algebra system, and when you do you will have to enter expressions through a typewriter, so it is well worth the trouble to get used to this notation. Indicate your understanding of the necessity to understand this notation.
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RESPONSE --> It is necessary to understand this notation because at some point in time, I will likely use this computer algebra system to transcribe problems/solutions. self critique assessment: 3
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21:38:01 `q005. At the link http://www.vhcc.edu/dsmith/genInfo/introductory problems/typewriter_notation_examples_with_links.htm (copy this path into the Address box of your Internet browser; alternatively use the path http://vhmthphy.vhcc.edu/ > General Information > Startup and Orientation (either scroll to bottom of page or click on Links to Supplemental Sites) > typewriter notation examples and you will find a page containing a number of additional exercises and/or examples of typewriter notation.Locate this site, click on a few of the links, and describe what you see there.
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RESPONSE --> At the listed webpage, there are about 30 or so examples of equations written in typewriter notation. When I click on the ""picture"" link, I am shown the same equation written in standard notation. confidence assessment: 3
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21:38:26 You should see a brief set of instructions and over 30 numbered examples. If you click on the word Picture you will see the standard-notation format of the expression. The link entitled Examples and Pictures, located in the initial instructions, shows all the examples and pictures without requiring you to click on the links. There is also a file which includes explanations. The instructions include a note indicating that Liberal Arts Mathematics students don't need a deep understanding of the notation, Mth 173-4 and University Physics students need a very good understanding,
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RESPONSE --> Agreed self critique assessment: 3
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21:38:54 while students in other courses should understand the notation and should understand the more basic simplifications. There is also a link to a page with pictures only, to provide the opportunity to translated standard notation into typewriter notation.
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RESPONSE --> OK self critique assessment: 3
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